The Relationship Between Geometric Sequences and First Order Difference Equations

A geometric sequence with a common ratio may be defined by a first order difference equation.

The Equation

A geometric sequence where there is a common ratio of $a$ can be defined by a first order difference equation of the form:

$t_{n+1}=at_n$

Where $a$ is the common ratio
$a>1$ is an increasing sequence
$0<a<1$ is a decreasing sequence
$a<0$ is a sequence alternating between positives and negatives

First Order Difference Equations Defining Geometric Sequences

For a first order difference equation to define a geometric sequence, there must be a common ratio and no addition or subtraction of terms. For first order difference equations, the geometric common ratio $r$ is the pro numeral $a$ .

Example 1

Which of the following first order difference equations defines a geometric sequence?

a) $t_{n+1}=4t_n$ $t_1=5$

This first order difference equation does define an (increasing) geometric sequence as it has a common ratio of 4.

b) $t_{n+1}=-2t_n$ $t_1=6$

This first order difference equation does define a geometric sequence (that alternates between positive and negative numbers) as it has a common ratio of -2.

c) $t_{n+1}=3t_n+2$

This first order difference equation does not define a geometric sequence as there is an addition of 2.

Expressing Geometric Sequences as First Order Difference Equations

If we have the general form of a geometric sequence, we can use $a$ to find $t_1$ and $r$ to find $a$ .

Be careful! $a$ in the form of $t_n=a(r)^{n-1}$ refers to the first term and $a$ in $t_{n+1}=at_n$ refers to the common ratio.

Example 2

Express the geometric sequence $t_n=4(3)^{n-1}$ where $n=1, 2, 3, ...$ as a first order difference equation.

In $t_n=4(3)^{n-1}$ , the common ratio ( $r$ ) is 3 and the first term ( $a$ ) is 4.

Therefore, for the first order difference equation $t_{n+1}=at_n$ , $a=3$ and $t_1=4$ .

Now we just have to substitute into the equation.

$t_{n+1}=3t_n$ $t_1=4$

We can also express a geometric sequence as a first order difference equation if we only have terms. We just have to calculate the common ratio and find the first term.

Example 3

Express the following geometric sequence as a first order difference equation.

$8, 24, 72, 216, 648, ...$

We are told that it is a geometric sequence so we don’t need to test for a common ratio.

$r=\frac {24}{8}$
$r=3$

So our common ratio is 3.

We can see that the first term is 8.

Therefore, the first order difference equation is given by $t_{n+1}=3t_n$ $t_1=8$

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